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Theorem imass1 6091
Description: Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
Assertion
Ref Expression
imass1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Proof of Theorem imass1
StepHypRef Expression
1 ssres 5999 . . 3 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
2 rnss 5929 . . 3 ((𝐴𝐶) ⊆ (𝐵𝐶) → ran (𝐴𝐶) ⊆ ran (𝐵𝐶))
31, 2syl 17 . 2 (𝐴𝐵 → ran (𝐴𝐶) ⊆ ran (𝐵𝐶))
4 df-ima 5680 . 2 (𝐴𝐶) = ran (𝐴𝐶)
5 df-ima 5680 . 2 (𝐵𝐶) = ran (𝐵𝐶)
63, 4, 53sstr4g 4020 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3941  ran crn 5668  cres 5669  cima 5670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680
This theorem is referenced by:  predrelss  6329  vdwnnlem1  16933  dprdres  19946  imasnopn  23538  imasncld  23539  imasncls  23540  utoptop  24083  restutop  24086  ustuqtop3  24092  utopreg  24101  metustbl  24419  imadifxp  32326  esum2d  33610  eulerpartlemmf  33893  bj-imdirco  36571  brtrclfv2  43027  frege97d  43052  frege109d  43057  frege131d  43064  hess  43080  resimass  44488  setrecsss  47993
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